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Wave subspaces
We use a family of synthetic waves based on the soliton sech2 to study wave subspaces. For example, consider a wave that propagates along the 45 degree line in the x-y plane with a speed, V. The wave has a sin(x)sin(y) factor that constrains the wave to be zero at the boundaries. The left graphic is a snapshot of the synthetic wave at the time it first attains maximum height (t = 43). The right graphic is a contour map showing the times at which the amplitude of the wave reaches 50% of its maximum value. The wave simply sweeps across the domain and disappears. Because of the sin(x)sin(y) factor in its definition, the data set does not represent a pure traveling wave.
Overview
The wave subspace computes a two-dimensional "best" representation of wave motion. We can display this two-dimensional representation as a curve of dots in the plane. The small inset in the upper left of the image below shows such a representation with the large red dot marking the current frame. The left graphic shows the actual wave at t = 50, while the right graphic shows the wave reconstruction in the two-dimensional phase space.
The following image shows another way of displaying a wave subspace relative to the full data set. Here the two data sets are overlaid, with the original data set using the same color scheme but with partial transparency. The wave subspace is represented by black dots at its spatial points. This method of display more clearly shows the exact relative positions of the data set and its wave subspace reconstruction than side-by-side comparison of the previous image. This figure shows two overlays, one at t = 32 and the other at t = 43.
The wave subspace has lower amplitude at both times and leads the original data set as the wave is developing at t = 32. The wave subspace amplitude is negative ahead of the wave. This structure is more visible at other viewing angles. We could also modify the dot color to enhance the visibility of negative values. The wave has reached its center position at t = 43, and the wave subspace trajectory is entirely in the primary wave subspace direction.